论文标题
非平滑整合性理论
Non-Smooth Integrability Theory
论文作者
论文摘要
我们研究了一种方法是从需求函数的候选者中计算出效用函数的方法，该函数不是可区分的，而是局部Lipschitz。使用这种方法，我们为需求函数的候选人提供了两个新的必要条件，以成为需求函数。第一个涉及Slutsky Matrix，第二个是存在偏微分方程的凹面解决方案。此外，我们表明，与需求函数相对应的上半连续弱顺序是唯一的，并且该弱顺序由我们计算出的效用函数表示。我们将这些结果应用于计量经济学理论。首先，我们表明，在几个需求下，如果一系列需求函数相对于紧凑型收敛的度量收敛到某些功能，则极限也是需求函数。其次，在上述度量标准下，在任何紧凑的集合上具有均匀Lipschitz常数的需求函数的空间。第三，从需求函数到计算的效用函数的映射变得连续。我们还显示了点上融合拓扑的类似结果。
We study a method for calculating the utility function from a candidate of a demand function that is not differentiable, but is locally Lipschitz. Using this method, we obtain two new necessary and sufficient conditions for a candidate of a demand function to be a demand function. The first concerns the Slutsky matrix, and the second is the existence of a concave solution to a partial differential equation. Moreover, we show that the upper semi-continuous weak order that corresponds to the demand function is unique, and that this weak order is represented by our calculated utility function. We provide applications of these results to econometric theory. First, we show that, under several requirements, if a sequence of demand functions converges to some function with respect to the metric of compact convergence, then the limit is also a demand function. Second, the space of demand functions that have uniform Lipschitz constants on any compact set is compact under the above metric. Third, the mapping from a demand function to the calculated utility function becomes continuous. We also show a similar result on the topology of pointwise convergence.
